- #1
apervaiz
- 2
- 0
The torque contribution due to the uniaxial anisotropy is given by the equation below
[tex] \frac{\Gamma}{l_m K} = (2 \sin\theta \cos\theta)[\sin\phi e_x - \cos\phi e_y] (3)[/tex]
This contribution can be taken in the LLG equation to derive the LLG equation in polar coordinates
[tex] \frac{\partial n_m}{\partial t} + ( n_m \times \frac{\partial n_m}{\partial t})=\frac{1}{2}\Omega_K \frac{\Gamma}{l_m K} (9) [/tex]
where [itex]n_m= [r,\theta,\phi][/itex]. Since r is unity for magnetization a differential equation in the two angles should be possible which should have the form
[tex] \begin{bmatrix}
\theta \ ' \\
\phi \ ' \\
\end{bmatrix}
= \begin{bmatrix}
\theta \\
\phi \\
\end{bmatrix}
[/tex]
Now the result of this derivation is already given as
[tex] \begin{bmatrix}
\theta \ ' \\
\phi \ ' \\
\end{bmatrix}
= \begin{bmatrix}
\alpha \sin \theta \cos \theta \\
\cos \theta \\
\end{bmatrix}
[/tex]
I'm having a hard time deriving this result from (3) using (9). Could anyone help me with this?
here is the link of this paper
http://journals.aps.org/prb/abstract/10.1103/PhysRevB.62.570
[tex] \frac{\Gamma}{l_m K} = (2 \sin\theta \cos\theta)[\sin\phi e_x - \cos\phi e_y] (3)[/tex]
This contribution can be taken in the LLG equation to derive the LLG equation in polar coordinates
[tex] \frac{\partial n_m}{\partial t} + ( n_m \times \frac{\partial n_m}{\partial t})=\frac{1}{2}\Omega_K \frac{\Gamma}{l_m K} (9) [/tex]
where [itex]n_m= [r,\theta,\phi][/itex]. Since r is unity for magnetization a differential equation in the two angles should be possible which should have the form
[tex] \begin{bmatrix}
\theta \ ' \\
\phi \ ' \\
\end{bmatrix}
= \begin{bmatrix}
\theta \\
\phi \\
\end{bmatrix}
[/tex]
Now the result of this derivation is already given as
[tex] \begin{bmatrix}
\theta \ ' \\
\phi \ ' \\
\end{bmatrix}
= \begin{bmatrix}
\alpha \sin \theta \cos \theta \\
\cos \theta \\
\end{bmatrix}
[/tex]
I'm having a hard time deriving this result from (3) using (9). Could anyone help me with this?
here is the link of this paper
http://journals.aps.org/prb/abstract/10.1103/PhysRevB.62.570